Hartshorne says an invertible sheaf $\mathcal{L}$ on $X$ is very ample relative to a field $k$ if there is an immersion $i:X\rightarrow \mathbb{P}_k^n$ for some $n$ such that $\mathcal{L}\simeq i^*\mathcal{O}(1)$. He then states this is equivalent to $\mathcal{L}$ generated by global sections $s_0,...,s_m$ such that $\phi: X\rightarrow \mathbb{P}_k^n$ given by $\phi: x\mapsto [s_0(x),...,s_m(x)]$ is an immersion (we of course need that $(\mathcal{L},H^0(X,\mathcal{L}))$ is base point free).
Question: How are these two criteria equivalent?
Let $Y=\mathbb P^m_k = \operatorname{Proj} k[T_0, \dots, T_m]$. The sections $T_0, \dots, T_m$ generate $\mathcal O(1)$, and induce global sections $s_0 = i^*T_0, \dots, s_m=i^*T_m$ of $i^*\mathcal O(1)$. Checking on stalks shows these generate $\mathcal L$. More precisely, let $y=i(x)$. The stalk $\mathcal L_x=i^*\mathcal O(1)_x$ is $\mathcal O(1)_y \otimes_{\mathcal O_{Y,y}}\mathcal O_{X,x}$, thus it follows that the fibers $s_{i,x}$ generate $\mathcal L_x$.
Conversely you can check $s_i = \phi^*T_i$, and that $\mathcal L$ is therefore $\phi^*\mathcal O(1)$.